A multitude of decisions face participants in the real estate market--buying, selling, leasing, lending, investing, developing, remodeling, and a host of others. These decisions are made by comparing the value of each decision to its cost. The value is partly determined by the timing of the benefits that the participant expects to receive from the investment. These benefits are often in the form of cash flows to the participant. In this chapter we will examine how the timing of benefits (cash flows, f or example) affects the value and rates of return and thus the decision making process.

The principles of the time value of money developed in this chapter have many applications, ranging from setting up schedules for paying off a mortgage to making decisions about whether to develop a shopping center. In fact, none of the techniques us ed in real estate decision making is more important than an understanding of the concepts of compound interest and present value. These concepts will form the basis for all the material in later chapters.

Suppose you are considering the purchase of a parcel of vacant land. The seller is asking $50,000. You expect land values to increase at a rate of 10 percent per year. The problem of how much the land will be worth at the end of 3 years illustrates the concept of compound interest, which simply means that during any given period, interest is earned on the principal amount and on the interest previously earned. Or, as in this problem, each succeeding year the value of the land will increase at the expe cted 10 percent rate over the previous year's value. The previous year's value includes the initial $50,000 plus the prior year's increases.

In any problem using compound interest, there are four variables:

- Present amount
- Future amount (or amounts)
- Rate of return (also called the interest rate, the discount rate, and the required rate)
- Length of time

The compound interest equations simply relate the four variables. To solve a given problem, always ask what you know and don't know. You must be given information (or make assumptions) about any three to determine the fourth. Any of the four variabl es may be the unknown.

Now let's return to our problem. What do we know? We know the following:

- Present amount -- $50,000
- Rate of return -- 10 percent
- Length of time -- 3 years

What do we not know? The expected future value of the land is the unknown. The future value can be calculated as follows:

The initial (present) value is $50,000. The buyer expects the property to increase by 10 percent ($5,000) in value during the first year. The future value at the end of the first year is thus $55,000. In equation form, the future value FV is

- equation 3-1
- FV
_{i,n}= PV(1 + i)^{n} - where PV =present amount or value
- i = rate
- n = number of periods

Thus the future value at the end of year 1 is

- FV
_{10%,1 yr}= $50,000(1 + 0.1)^{1} - = $55,000

During the second year the buyer expects the land to increase in value by 10 percent over the $55,000, or $5,500. This results in a value of $60,500 at the end of year 2. In equation form,

- FV
_{10%,2 yr}= $50,000(1 + 0.1)^{2} - = $50,000(1 + 0.1)
- = $50,000(1.21)
- = $60,500

Repeating this process, the future value after 3 years is

- FV
_{10%,3 yr}= $50,000(1 + 0.1)^{3} - = $50,000(1 + .331)
- = $60,550

Thus if the investor bought the land for $50,000 and if it increased at a rate of 10 percent (compounded each year), the value would be $66,550 at the end of 3 years.

Equation 3-1 is used to find the future value of a known present amount when the future amount will be received in a lump sum at a particular time n.

To simplify calculations, tables have been developed that depict the future value of $1.00 for various rates and time periods. The tables simplify Equation 3-1 in that the (1 + i)n calculation has already been done. All that remains is to multiply t he present value (in dollars) by the appropriate factor from the table (see Table 3-1). Equation 3-1 can thus be expressed as

- FV
_{i,n}= PV(FVF_{i,n})

where FVF_{i,n} = the future value factor at rate i for n periods, or (1 + i)^{n}. Notice that FVF depends only on the rate and the length of time.

Returning again to our example, the future value factor is

- FV
_{10%,3 yr}= $50,000(1.331) - = $66,550

which is identical to the value obtained previously. See Sample Problem 3-1 for another example of the computation of the future value of a lump sum.

A real estate investor is considering the purchase of an apartment building. The investor estimates the building's current value at $400,000. At an annual increase of 8 percent, how much would the investment be worth at the end of 5 years? To solve this problem, first recall Equation 3- l:

- FV
_{i,n}= PV(1 + i)^{n}

We know the following

- PV = $400,000
- i = 0.08
- n = 5

Substituting the knowns into the equation,

- FV
_{8%,5 yr}= $400,000(1 + .08)^{5} - = $400,000(1.46933)
- = $587,731